Dynamos driven by helical waves: scaling laws for numerical dynamos and for the planets

Abstract

We derive scaling relationships for planetary dynamos based on a balance between energy production and Joule dissipation, and between the curl of the buoyancy and Coriolis forces. These scaling relationships are deduced for the particular case of dynamos driven by helical waves, but are shown to have a much broader applicability. They are consistent with the evidence of the numerical dynamos, yielding predictions consistent with published empirical scaling laws and also with the observed transition from dipolar to multipolar dynamos. A direct comparison with the observational evidence for the planets is hampered by the fact that we do not know what sets the smallest scale of the motion in the planets. Nevertheless, we use our scaling relationships to show that the traditional assumption that the Elsasser number is of order unity is inconsistent with the observation that the gas-giant dynamos are dipolar dynamos, as is the more recent suggestion that the strength of the dipole is independent of rotation rate and controlled by the buoyancy flux alone. On the other hand, we show that the observational data is consistent with the hypothesis that a dipolar dynamo saturates at the lowest permissible magnetic energy compatible with a given buoyancy flux.